A ride-sharing car moving along a straight section of road starts from rest, accelerating at 2.00 m/s2 until it reaches a speed of 34.0 m/s. Then the vehicle moves for 37.0 s at constant speed until the brakes are applied, stopping the vehicle in a uniform manner in an additional 5.00 s.

(a)

How long is the ride-sharing car in motion (in s)?

 s

(b)

What is the average velocity of the ride-sharing car for the motion described? (Enter the magnitude in m/s.)

 m/s

To solve this problem, we can break it down into three distinct phases:

1. **Acceleration phase**: The car accelerates from rest.
2. **Constant speed phase**: The car moves at a constant speed.
3. **Deceleration phase**: The car slows down uniformly to a stop.

Let's go through each part step by step.

### (a) Total Time in Motion

#### Phase 1: Acceleration
- Initial velocity: \( u_1 = 0 \, \text{m/s} \)
- Final velocity: \( v_1 = 34.0 \, \text{m/s} \)
- Acceleration: \( a_1 = 2.00 \, \text{m/s}^2 \)

We use the first kinematic equation to find the time taken to reach the speed of 34.0 m/s:
\[
v_1 = u_1 + a_1 t_1
\]
Substituting the known values:
\[
34.0 = 0 + 2.00 t_1
\]
\[
t_1 = \frac{34.0}{2.00} = 17.0 \, \text{s}
\]

#### Phase 2: Constant Speed
- Time at constant speed: \( t_2 = 37.0 \, \text{s} \)

#### Phase 3: Deceleration
- Initial velocity: \( u_3 = 34.0 \, \text{m/s} \)
- Final velocity: \( v_3 = 0 \, \text{m/s} \)
- Time to stop: \( t_3 = 5.00 \, \text{s} \)

We use the same kinematic equation as in Phase 1 to find the acceleration:
\[
v_3 = u_3 + a_3 t_3
\]
Here, \( v_3 = 0 \) (since the car stops), so:
\[
0 = 34.0 + a_3 \times 5.00
\]
\[
a_3 = \frac{-34.0}{5.00} = -6.80 \, \text{m/s}^2
\]
The deceleration is \( -6.80 \, \text{m/s}^2 \), but the time is already given as 5.00 s.

Now, sum up the times from all three phases:
\[
\text{Total time} = t_1 + t_2 + t_3 = 17.0 \, \text{s} + 37.0 \, \text{s} + 5.00 \, \text{s} = 59.0 \, \text{s}
\]

### Final Answer for (a):
(a) The ride-sharing car is in motion for **59.0 seconds**.

### (b) Average Velocity

The average velocity is calculated as the total displacement divided by the total time in motion.

#### Phase 1: Displacement during Acceleration
Use the kinematic equation:
\[
d_1 = u_1 t_1 + \frac{1}{2} a_1 t_1^2
\]
Substituting the known values:
\[
d_1 = 0 \times 17.0 + \frac{1}{2} \times 2.00 \times (17.0)^2 = \frac{1}{2} \times 2.00 \times 289 = 289.0 \, \text{m}
\]

#### Phase 2: Displacement during Constant Speed
Since the car is moving at a constant speed:
\[
d_2 = v_1 \times t_2 = 34.0 \times 37.0 = 1258.0 \, \text{m}
\]

#### Phase 3: Displacement during Deceleration
Use the kinematic equation again:
\[
d_3 = u_3 t_3 + \frac{1}{2} a_3 t_3^2
\]
Substituting the known values:
\[
d_3 = 34.0 \times 5.00 + \frac{1}{2} \times (-6.80) \times (5.00)^2
\]
\[
d_3 = 170.0 - \frac{1}{2} \times 6.80 \times 25.0 = 170.0 - 85.0 = 85.0 \, \text{m}
\]

#### Total Displacement
\[
\text{Total displacement} = d_1 + d_2 + d_3 = 289.0 + 1258.0 + 85.0 = 1632.0 \, \text{m}
\]

#### Average Velocity
\[
\text{Average velocity} = \frac{\text{Total displacement}}{\text{Total time}} = \frac{1632.0 \, \text{m}}{59.0 \, \text{s}} \approx 27.66 \, \text{m/s}
\]

### Final Answer for (b):
(b) The average velocity of the ride-sharing car is **27.66 m/s**.

Would you like more details or have any questions about the solution?

Here are some related questions you might find useful:

1. What would be the final velocity if the car decelerated over a longer period, say 7 seconds?
2. How much time would the car spend in motion if the initial acceleration was higher, like 3 m/s²?
3. What would be the total displacement if the constant speed phase lasted 50 seconds instead of 37 seconds?
4. How would the average velocity change if the car reached a higher final speed of 40 m/s?
5. What if the car was moving uphill during the constant speed phase, requiring additional force? How would this affect the results?

**Tip:** When calculating average velocity over multiple phases of motion, always ensure you account for both time and displacement accurately in each phase.

Explore how a ride-sharing car accelerates, moves at constant speed, and decelerates uniformly. Calculate the total time in motion and average velocity using kinematic equations. Suitable for advanced high school students.

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